Spare parts inventory pooling games

Transcription

1 School of Industrial Engineering Eindhoven University of Technology International Workshop on Supply Chain s for Shared Resource Management, Brussels, 2010

2 Outline Introduction 1 Introduction 2 3 4

3 Spare parts Introduction Equipment-intensive high-tech industries Example: airlines, nuclear power plants High availability needed Random (infrequent) failures of critical components Can cause the machine to go down Spare parts are kept on stock To combat costly downtimes Typically very expensive with low demand rates

4 Spare parts inventory pooling Inventory pooling: Demand at a stockpoint that is out of stock is satisfied from another stockpoint This can improve availability and reduce costs Stockpoint with spare parts Stockpoint with spare parts Demands Demands Group of machines Group of machines

5 Situation: Several independent companies that separately stock spare parts; may reduce expected costs by pooling inventory Problem: Joint costs have to be allocated amongst the participating companies in a pooling group. Can this be done in a fair, stable way? Is the core of the associated cooperative game non-empty?

6 Situation: Several independent companies that separately stock spare parts; may reduce expected costs by pooling inventory Problem: Joint costs have to be allocated amongst the participating companies in a pooling group. Can this be done in a fair, stable way? Is the core of the associated cooperative game non-empty?

7 Cooperative game theory and the core Player set N = {1, 2,..., n} Cost function c : 2 N R So coalition M N alone faces costs c(m) Cooperative cost game: (N, c) Allocation (x i ) i N R N distributes c(n) over all players. Definition Core: set of all allocations that are Efficient: i N x i = c(n) Stable: for all M N, i M x i c(m)

8 Cooperative game theory and the core Player set N = {1, 2,..., n} Cost function c : 2 N R So coalition M N alone faces costs c(m) Cooperative cost game: (N, c) Allocation (x i ) i N R N distributes c(n) over all players. Definition Core: set of all allocations that are Efficient: i N x i = c(n) Stable: for all M N, i M x i c(m)

9 The research question Does a spare parts inventory pooling game have a non-empty core?

10 Literature overview Introduction Related topics: Spare parts inventory pooling Axsäter (1990), Alfredsson and Verrijdt (1999), Grahovac and Chakravarty (2001), Paterson et al. (2009)... Cooperative inventory games Hartman and Dror (1996), Slikker et al. (2005), Anily and Haviv (2007), Özen et al. (2008),... Zhao et al. (2005, 2006), Wong et al. (2007), Kilpi et al. (2009).

11 The base setting - notation Base setting: companies (almost) identical (symmetry assumptions will be relaxed later) Set of companies: N = {1, 2,..., n} Demands: Poisson(λ) Repairs: Expected lead time 1/µ Stock-out: emergency procedure; costs c em Base stock level: S Holding cost rate: h i 1/µ 1/µ S parts h 1... S parts h n c em c em Poi(λ) Pooling within coalition Poi(λ)

12 Cooperation between companies c em Assumptions: Negligible distance between stockpoints Full pooling is applied Each company is interested in the expected infinite horizon costs per time unit. The chosen base stock level is fixed 1/µ 1/µ S parts h 1... S parts h n c em Poi(λ) Pooling within coalition Poi(λ)

13 Stock-out probability and cost function Consider coalition M N, i.e. m = M companies. Steady-state probability of having 0 parts on hand in the coalition: Erlang loss function: π 0 (ms, ρ) = ρms /(ms)! mλ ms with ρ = y=0 ρy /y! µ Cost function: c(m) = i M h is + π 0 (ms, ρ) mλ c em 1/µ 1/µ S parts h 1... S parts h n c em c em Poi(λ) Pooling within coalition Poi(λ)

14 Stock-out probability and cost function Consider coalition M N, i.e. m = M companies. Steady-state probability of having 0 parts on hand in the coalition: Erlang loss function: π 0 (ms, ρ) = ρms /(ms)! mλ ms with ρ = y=0 ρy /y! µ Cost function: c(m) = i M h is + π 0 (ms, ρ) mλ c em 1/µ 1/µ S parts h 1... S parts h n c em c em Poi(λ) Pooling within coalition Poi(λ)

15 An example game Introduction Example N = {1, 2, 3}, S = 1, λ = 5, µ = 25, h i = 4000, c em = π 0 (1, 5 25 ) 0.17, π 0(2, ) 0.05, π 0(3, ) Coalition size M Characteristic costs c(m) (rounded) Core ; e.g. allocation x i = 5300 i N is a core element. But the game is not concave!

16 An example game Introduction Example N = {1, 2, 3}, S = 1, λ = 5, µ = 25, h i = 4000, c em = π 0 (1, 5 25 ) 0.17, π 0(2, ) 0.05, π 0(3, ) Coalition size M Characteristic costs c(m) (rounded) Core ; e.g. allocation x i = 5300 i N is a core element. But the game is not concave!

17 Outline of the analysis 1 st extension Allow for asymmetric emergency costs 3 rd extension Base setting: n (almost) identical companies Combination 2 nd extension Allow for asymmetric base stock levels and demand rates 3 rd extension

18 Base Setting Introduction Theorem Proof. Situation: (N, S, λ, µ, (h i ) i N, c em ). Core of associated game is non-empty. Consider a cost allocation ( with ) No Transfer Payments: NTP i = h i S + π 0 N S, N λ λc em, i N NTP Core(N, c). µ

19 Asymmetric Emergency Costs Theorem Proof. Situation: ( N, S, λ, µ, (h i ) i N, (c em i ) i N ). Core of associated game is non-empty. Consider a cost allocation ( with ) No Transfer Payments: NTPi E = h i S + π 0 N S, N λ λci em, i N NTP E Core(N, c). µ

20 Asymmetric Demand Rates and Base Stock Levels Theorem Situation: (N, (S i ) i N, (λ i ) i N, µ, (h i ) i N, c em ) c(m) = ( h i S i + π 0 S i, ) λ i λ i c em, M N µ i M i M i M i M Core of associated game is non-empty.

21 Asymmetric Demand Rates and Base Stock Levels Proof. For a setting with identical λ but allowing asymmetric S: Use convexity and subadditivity properties of the Erlang loss function = Shapley-Bondareva conditions hold A game in such a setting has a non-empty core. Take a setting allowing asymmetric λ and S. Split each company into sub-companies to construct a game with identical λ but allowing asymmetric S. Take a core element from this constructed game. Transform it into a core element for the setting allowing asymmetric λ and S.

22 Combination of asymmetries Counterexamples with empty cores for situations with Asymmetric emergency costs and base stock levels Asymmetric emergency costs and demand rates Main problem: full pooling approach.

23 Intuition: Non-optimal Full Pooling high c em 1 part low c em 0 parts

24 Intuition: Non-optimal Full Pooling high c em 1 part low c em Pooling? 0 parts No pooling? Demand

25 Intuition: Non-optimal Full Pooling Pooling high c em high c em 1 part 0 part Pooling? low c em low c em 0 parts No pooling? Demand 0 parts

26 Intuition: Non-optimal Full Pooling Pooling No pooling high c em high c em high c em 1 part 0 part 1 part low c em Pooling? low c em low c em 0 parts No pooling? Demand 0 parts 0 parts

27 Introduction Setting: several independent high-tech companies that stock repairable spare parts of the same item can pool their inventory Main results: Core For the base setting and for settings allowing non-identical emergency costs. non-identical base stock levels and demand rates; or But not always for a combination of asymmetries! Future/current research: extensions Non-zero lateral transshipment costs, optimized base stock levels, partial pooling approach, two-echelon structure,...